Programming Language Semantics Denotational Semantics

1
Programming Language SemanticsDenotational
Semantics

• Chapter 5

2
Outline

• Least fixed points on relations
• Motivation
• Denotational Semantics
• Least fixed points
• Equivalence of Semantics
• Complete partial orders and continuous functions

3
Operators and their least fixed points

• For a set of rule instances R
• R(B)y ?X ?B, X/y ? R
• Proposition 4.11 A set B is closed under R if
  R(B) ?B
• R is monotonic
• A ? B ? R(A) ? R(B)
• Define the sequence of sets
• A0 R0(?) ?
• A1 R1(?) R(?)
• A2 R2(?) R(R(?))
• An Rn(?)
• Define A ?n ?? An

4
Proposition 4.12

1. A is R-closed
2. R(A) A
3. A is the least R-closed set

Alternative Formulation
Let fix(R) denotes the least fixed point of R

1. R(fix(R)) fix(R)
2. ?X. R(X) X ? fix(R) ?X

Claim fix(R) ?n?? Rn(?)A
5
Motivation for Denotational Semantics

• Equivalence of commands
• c1 ? c2 ? ??,??? ltc1, ?gt ? ? ? ltc2, ?gt ? ?
• The meaning of a command is a (partial) state
  transformer
• C? c1 ? (?,?) ltc1, ?gt ? ?
• C? c2 ? (?,?) ltc2, ?gt ? ?
• c1 ? c2 ? C? c1 ? C? c2 ?
• Denotational semantics
• A?a ? ??N
• B?b ? ??T
• C?c ? ???

6
Denotational semantics

• A Aexp ? (??N)
• B Bexp ?(??T)
• C Com ?(???)
• Defined by structural induction

7
Denotational semantics of Aexp

• A Aexp ? (??N)
• A ?n? (?, n) ? ??
• A ?X? (?, ?(X)) ? ??
• A ?a0a1? (?, n0n1) (?, n0)?A?a0?,
  (?,n1)?A?a1?
• A ?a0-a1? (?, n0-n1) (?, n0)?A?a0?,
  (?,n1)?A?a1?
• A ?a0?a1? (?, n0 ? n1) (?, n0)?A?a0?,
  (?,n1)?A?a1?

Lemma A ?a ? is a function
8
Denotational semantics of Aexp with ?

• A Aexp ? (??N)
• A ?n? ????.n
• A ?X? ????.?(X)
• A ?a0a1? ????.(?a0??A?a1??)
• A ?a0-a1? ????.(?a0??-A?a1??)
• A ?a0?a1? ????.(?a0?? ? A?a1??)

9
Denotational semantics of Bexp

• B Bexp ? (??T)
• B ?true? (?, true) ? ??
• B ?false? (?, false) ? ??
• B ?a0a1? (?, true) ? ?? A?a0??A?a1? ??
  (?, false) ? ??
  A?a0???A?a1? ?
• B ?a0?a1? (?, true) ? ?? A?a0?? ? A?a1?
  ?? (?, false) ? ??
  A?a0???A?a1? ?
• B ??b? (?, ?T t) ? ??, (?, t) ?B?b?
• B ?b0?b1? (?, t0 ?Tt1) ? ??, (?, t0) ?B?b0?,
  (?, t1) ?B?b1?
• B ?b0?b1? (?, t0 ?Tt1) ? ??, (?, t0) ?B?b0?,
  (?, t1) ?B?b1?

Lemma B?b? is a function
10
Denotational semantics of Com

• C Com ? (???)
• C ?skip? (?, ?) ? ??
• C ?Xa? (?, ?n/X) ? ?? nA?a??
• C ?c0c1? C ?c1? ? C ?c0?
• C ?if b then c0 else c1? (?, ?) B?b??true
  (?, ?)? C?c0??(?, ?) B?b??false (?,
  ?)? C?c1?
• C ?while b do c? ?

11
Denotational semantics of Com

• C Com ? (???)
• C ?if b then c0 else c1? (?, ?) B?b??true
  (?, ?)? C?c0??(?, ?) B?b??false (?,
  ?)? C?c1?
• w while b do c
• w ? if b then cw else skip
• C ?w? (?, ?) B?b??true (?, ?)?
  C?cw??(?, ?) B?b??false (?, ?)
  B?b??true (?, ?)? C?w??C?c? ?(?, ?)
  B?b??false
• ? (?, ?) ??(?)true (?, ?)? ? ? ?
  ?(?, ?) ??(?) false

12
Denotational semantics of Com

• C Com ? (???)
• ? (?, ?) ??(?)true (?, ?)? ? ? ?
  ?(?, ?) ??(?) false
• ? (?) (?, ?) ??(?)true (?, ?)? ? ? ?
  ?(?, ?) ??(?) false(?, ?) ?? ?
  ?(?)true (?, ?)? ?, (?, ?)? ? ?(?, ?)
  ??(?) false
• ? (?) ?

13
Denotational semantics of Com

• C Com ? (???)
• ? (?) (?, ?) ??(?)true (?, ?)? ? ? ?
  ?(?, ?) ??(?) false(?, ?) ?? ?
  ?(?)true (?, ?)? ?, (?, ?)? ? ?(?, ?)
  ??(?) false
• ? (?) ?
• ? R
• R((?, ?)/ (?, ?)) ?(?)true (?, ?)?
  ? ? (?/ (?, ?)) ??(?) false
• ? fix(R)

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Denotational semantics of Com

• C?skip?(?, ?) ???
• C ?Xa? (?, ?n/X) ? ?? nA?a??
• C ?c0c1? C ?c1? ? C ?c0?
• C ?if b then c0 else c1? (?, ?) B?b??true
  (?, ?)? C?c0??(?, ?) B?b??false (?,
  ?)? C?c1?
• C ?while b do c? fix(?) where?(?) (?, ?)
  B?b??true (?, ?)? ? ?C?c? ?(?, ?) ?
  B?b??false

15
Proposition 5.1
w while b do c C ?w? C ?if b then cw else
skip?
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Equivalence of Semantics

• Lemma 5.3 For all a ? Aexp
• A?a?(?, n) lt ?, agt? n

17
Equivalence of Semantics

• Lemma 5.4 For all b ? Bexp
• B?b?(?, t) lt ?, bgt? t

18
Equivalence of Semantics

• Lemma 5.6 For all c ? Com
• lt?, cgt? ? ? ( ?, ?) ? C?c?
• Theorem 5.7 For all c ? Com
• C?c? ( ?, ?) lt?, cgt? ?

19
Partial Orders

• A partial order (p.o.) is a set P with a binary
  relation ?
• reflexive ?p p ?P. p?p
• transitive ?p, q, r ?P. p ?q q ?r ?p ?r
• antisymmetric ?p, q ?P. p ?q q ?p ?pq
• For a partial order (P, ?) and a subset X?P
• p is an upper bound of X
• ?q ?X. q ?p
• p is a least upper bound of X (denoted by ?X) if
• p is an upper bound of X
• For all upper bounds q of X p ?q

d1 ? d2 ? dn ?d1, d2, , dn
20
Complete Partial Orders

• Let (D, ?) be a partial order
• An ?-chain is an increasing chain
• d0 ?d1 ?dn ?..
• D is a complete partial order (c.p.o) if every
  ?-chain has a least upper bound
• D is a complete partial order with ? if is a
  c.p.o. with a minimum element ?

21
Monotonic and Continus Functions

• A function f D?E between cpos D and E is
  monotonic if
• ?d,d ?D.d ?d ?f(d) ?f(d)
• Such a function is continuous if for all chains
  d0 ?d1 ?dn ? in D ?n?? f(dn) f(? n?? dn)

22
Fixed Points

• Let f D ?D be a continuous function on a cpo
  with ? D
• A fixed point of f is an element d ?D such that
  f(d) d
• A pre fixed point of f is an element d ?D such
  that f(d) ?d
• Thm fix(f) ?n?? fn(?)

23
Missing

• Greatest lower bounds
• Knaster-Tarski Theorem
• Conclusion